Inequality Application

**mathgeek777** · August 22nd 2008, 12:14 PM

A charter airline finds that on its Saturday flights from Philadelphia to London, all 120 seats will be sold if the ticket price is $200. However, for each $3 increase in ticket price, the number of seats sold decreases by one.

a) Find a formula for the number of seats sold if the ticket price is P dollars.

The answer for a is $-\frac{1}{3}P + \frac{560}{3}$ . However, I do not see how they arrived at that answer. Any help will be greatly appreciated.

**Jhevon** · August 22nd 2008, 12:28 PM

Originally Posted by mathgeek777

The answer for a is $-\frac{1}{3}P + \frac{560}{3}$ . However, I do not see how they arrived at that answer. Any help will be greatly appreciated.

do you realize that we have a linear relationship between the number of seats filled and the ticket price? since each changes at a constant rate with respect to each other.

lets think of this graphically so that you get the idea. we want a function of price, so put that on the x-axis. the number of seats on the y-axis. now, plot a few points:

$\begin{array}{c|c} S & P \\ \hline 120 & 200 \\ 119 & 203 \\ 118 & 206 \\ . & . \\ . & . \\ . & . \end{array}$

S is the number of seats, P is the price. S goes down by 1 when P goes up by 3

all we need to do is find the straight line that passes through these points. can you do that?

**mathgeek777** · August 22nd 2008, 01:37 PM

Ok. I see where the $-\frac{1}{3}$ came from, but i still don't know how you get $\frac{560}{3)$

**Jhevon** · August 22nd 2008, 01:53 PM

Originally Posted by mathgeek777

Ok. I see where the $-\frac{1}{3}$ came from, but i still don't know how you get $\frac{560}{3}$

the equation of a line in the slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ , is the y-intercept.

given any two points on the line, $(x_1, y_1)$ and $(x_2,y_2)$ , we can find the equation of the line as follows:

$m = \frac {y_2 - y_1}{x_2 - x_1}$

By the point-slope form, plug in the values of $m$ , $x_1$ and $y_1$ into:

$y - y_1 = m(x - x_1)$

and solve for $y$ to get it into the slope-intercept form.

alternatively, we could plug the points into $y = mx + b$ and solve for $b$ , since it would be the only unknown at this point

here of course, your y is S and your x is P

**Soroban** · August 22nd 2008, 01:58 PM

Hello, mathgeek777!

A charter airline finds that on its Saturday flights from Philadelphia to London,
all 120 seats will be sold if the ticket price is $200.
However, for each $3 increase in ticket price, the number of seats sold decreases by one.

a) Find a formula for the number of seats sold if the ticket price is $P$ dollars.

The answer is: $-\frac{1}{3}P + \frac{560}{3}$

Let $x$ = number of $3-increases in the price of a ticket.

Then the price is: . $P \:=\:200 + 3<br /> x\quad\Rightarrow\quad x \:=\:\frac{P - 200}{3}\;\;{\color{blue}[1]}$

. . and the number of seats sold is: . $S \;=\;120 - x\;\;{\color{blue}[2]}$

Substitute [1] into [2]: . $S \;=\;120 - \frac{P-200}{3} \;=\;\frac{360}{3} - \frac{P-200}{3} \;=\;\frac{-P + 560}{3}$

Therefore: . $S \;=\;-\frac{1}{3}P + \frac{560}{3}$

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